The former striker also says qualifying for the Champions League is a step in the right direction.
The Red Devils are expected to be busy during the summer transfer window as manager Louis Van Gaal looks to revamp his squad, with a number of players expected to arrive for big money while several will also be leaving.
They were obviously busy last summer as well, with the signing of Angel Di Maria the most costly of several acquisitions, and Cole believes they need to compete for the best players if they want to challenge in the Premier League and on European football's grandest stages.
“Manchester United need to be competing for the best players in the world if they want to challenge in the Premier League and in Europe next season,” he said.
“If you are going to compete in the Champions League, look at the final you’ve had; Barcelona against Juventus. Look at the guys those clubs sign. The calibre of players you have to be looking at is what they have got and United have to be fishing in the same pool as those clubs if they want to improve.”
It was a relatively tough season for Dutchman Van Gaal in the Premier League last term, his first in England, with his side initially struggling as he failed to find a formation and line-up that suited him.
United did, however, rally in the second half of the campaign, with a fine run of form having helped them to secure a fourth-placed finish and a return to the Champions League at the first time of asking.
And Cole believes qualification for Europe is obvious progress from last season.
“It’s been another tough season for United, but finishing in the Champions League spot is naturally progress,” he added.
“The form has been up and down but a club like Manchester United needs to be in the Champions League to attract the best players and make progress for next season, to hopefully try and compete for the Premier League.”